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Hilbert modular forms and class numbers

Qinyun Tan, Bingyong Xie

Abstract

In 1975, Goldfeld gave an effective solution to Gauss's conjecture on the class numbers of imaginary quadratic fields. In this paper, we generalize Goldfeld's theorem to the setting of totally real number fields.

Hilbert modular forms and class numbers

Abstract

In 1975, Goldfeld gave an effective solution to Gauss's conjecture on the class numbers of imaginary quadratic fields. In this paper, we generalize Goldfeld's theorem to the setting of totally real number fields.
Paper Structure (13 sections, 41 theorems, 191 equations)

This paper contains 13 sections, 41 theorems, 191 equations.

Table of Contents

  1. Quadratic forms on $\mathfrak{o}$-modules
  2. Definite quadratic forms
  3. Fundamental quadratic forms
  4. Equivalence classes of fundamental quadratic forms
  5. An estimate on $h_K$
  6. An estimate on $h$
  7. Hilbert modular form and Dirichlet series
  8. Hilbert modular forms
  9. Certain Dirichlet series
  10. Some estimations
  11. Number of lattice points
  12. Measures and their Mellin transforms
  13. Proof of Theorem \ref{['thm:main0']}

Key Result

Theorem 2

$($Hecke-Deuring-Heilbronn LandauMD33H$)$ Let $K$ run through all imaginary quadratic fields. If $D_K\rightarrow \infty$, then $h(K)\rightarrow \infty$.

Theorems & Definitions (85)

  • Conjecture 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Lemma 1.1
  • proof
  • Lemma 1.2
  • proof
  • Lemma 1.3
  • ...and 75 more